Question 11 Mark
Represent the following rational numbers on the number line:
$\frac{8}{3}$
$\frac{8}{3}$
Answer
View full question & answer→$\frac{8}{3}=2\frac{2}{3}$
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Question 21 Mark
Simplify:
$(3^4)^{\frac{1}{4}}$
$(3^4)^{\frac{1}{4}}$
Answer
View full question & answer→$(3^4)^{\frac{1}{4}}=3^{\big(4\times\frac{1}{4}\big)}=(3)^1=3$
Question 31 Mark
Classify the following number as rational or irrational. give reasons to support your answer.
$\sqrt{\frac{3}{81}}$
$\sqrt{\frac{3}{81}}$
Answer
View full question & answer→$\sqrt{\frac{3}{81}}$
$\sqrt{\frac{3}{81}}=\sqrt{\frac{1}{27}}=\frac{1}{3}\sqrt{\frac{1}{3}}$
It is an irrational number.
$\sqrt{\frac{3}{81}}=\sqrt{\frac{1}{27}}=\frac{1}{3}\sqrt{\frac{1}{3}}$
It is an irrational number.
Question 41 Mark
Classify the following number as rational or irrational. give reasons to support your answer.
4.1276
4.1276
Answer
View full question & answer→4.1276
It is a terminating decimal. Hence, it is rational.
It is a terminating decimal. Hence, it is rational.
Question 51 Mark
Classify the following number as rational or irrational. give reasons to support your answer.
6.834837...
6.834837...
Answer
View full question & answer→6.834837... It is neither terminating, nor repeating hence it is irrational number.
Question 61 Mark
Without actual division, find the following rational numbers are terminating decimals. $\frac{31}{375}$
Answer
View full question & answer→$\frac{31}{375}$
Denominator of $\frac{31}{375}$ is $375.$
$375 = 5^3 \times 3$
So, the prime factor $375$ are $5$ and $3$.
Thus$, \frac{31}{375}$ is not a terminating decimal.
Denominator of $\frac{31}{375}$ is $375.$
$375 = 5^3 \times 3$
So, the prime factor $375$ are $5$ and $3$.
Thus$, \frac{31}{375}$ is not a terminating decimal.
Question 71 Mark
Evaluate:
$\big(125\big)^{\frac{1}{3}}$
$\big(125\big)^{\frac{1}{3}}$
Answer
View full question & answer→$\big(125\big)^{\frac{1}{3}}=(5^3)^{\frac{1}{3}}=5^{3\times\frac{1}{3}}=5^1=5$
Question 81 Mark
Classify the following number as rational or irrational. give reasons to support your answer.
3.040040004...
3.040040004...
Answer
View full question & answer→3.040040004... is an irrational number because it is a non-terminating, non-repeating decimal.
Question 91 Mark
Rationalise the denominator of the following:
$\frac{1}{\sqrt{7}}$
$\frac{1}{\sqrt{7}}$
Answer
View full question & answer→On multiplying the numerator and denominator of the given number by $\sqrt{7},$ we get
$\frac{1}{\sqrt{7}}=\frac{1}{\sqrt{7}}\times\frac{\sqrt{7}}{\sqrt{7}}=\frac{\sqrt{7}}{7}.$
$\frac{1}{\sqrt{7}}=\frac{1}{\sqrt{7}}\times\frac{\sqrt{7}}{\sqrt{7}}=\frac{\sqrt{7}}{7}.$
Question 101 Mark
Is zero a rational number? Justify.
Answer
View full question & answer→Yes, 0 is a rational number.
0 can be expressed in the form of the fraction $\frac{\text{p}}{\text{q}},$ where p = 0 and q can be any integer except 0.
0 can be expressed in the form of the fraction $\frac{\text{p}}{\text{q}},$ where p = 0 and q can be any integer except 0.
Question 111 Mark
Simplify $\big(2\sqrt{5}+3\sqrt{2}\big)^2.$
Answer
View full question & answer→$\big(2\sqrt{5}+3\sqrt{2}\big)^2$
$=\big(2\sqrt{5}\big)^2+2\times2\sqrt{5}\times3\sqrt{2}+\big(3\sqrt{2}\big)^2$
$=20+12\sqrt{10}+18$
$=38+12\sqrt{10}$
$=\big(2\sqrt{5}\big)^2+2\times2\sqrt{5}\times3\sqrt{2}+\big(3\sqrt{2}\big)^2$
$=20+12\sqrt{10}+18$
$=38+12\sqrt{10}$
Question 121 Mark
Give an example of two irrational numbers whose:
Quotient is a rational number.
Quotient is a rational number.
Answer
View full question & answer→2 irrational numbers with quotient a rational number will be $\sqrt{63}$ and $\sqrt{7}$
Question 131 Mark
Classify the following number as rational or irrational. give reasons to support your answer.
$\frac{22}{7}$
$\frac{22}{7}$
Answer
View full question & answer→$\frac{22}{7}$ is a rational number because it can be expressed in the $\frac{\text{p}}{\text{q}}$ form.
Question 141 Mark
Without actual division, find the following rational numbers are terminating decimals.
$\frac{16}{125}$
$\frac{16}{125}$
Answer
View full question & answer→$\frac{16}{125}$
Denominator of $\frac{16}{125}$ is $125$.
And$, 125 = 5^3$
Therefore$, 125$ has no other factors than $2$ and $5.$
Thus$, \frac{16}{125}$ is a terminating decimal.
Denominator of $\frac{16}{125}$ is $125$.
And$, 125 = 5^3$
Therefore$, 125$ has no other factors than $2$ and $5.$
Thus$, \frac{16}{125}$ is a terminating decimal.
Question 151 Mark
Give an example of two irrational numbers whose:
Product is a rational number.
Product is a rational number.
Answer
View full question & answer→2 irrational numbers with product a rational number will be $5+\sqrt{7}$ and $5-\sqrt{7}$
Question 161 Mark
Simplify:
$6^\frac{1}{2}\times7^\frac{1}{2}$
$6^\frac{1}{2}\times7^\frac{1}{2}$
Answer
View full question & answer→$6^\frac{1}{2}\times7^\frac{1}{2}=(6\times7)^{\frac{1}{2}}=(42)^{\frac{1}{2}}$
Question 171 Mark
Represent the following rational numbers on the number line:
$\frac{5}{7}$
$\frac{5}{7}$
Answer
View full question & answer→$\frac{5}{7}$
Question 181 Mark
Simplify:
$3^\frac{1}{4}\times5^\frac{1}{4}$
$3^\frac{1}{4}\times5^\frac{1}{4}$
Answer
View full question & answer→$3^\frac{1}{4}\times5^\frac{1}{4}=(3\times5)^{\frac{1}{4}}=(15)^{\frac{1}{4}}$
Question 191 Mark
Give an example of two irrational numbers whose:
Difference is a rational number.
Difference is a rational number.
Answer
View full question & answer→2 irrational numbers with difference is a rational number will be $5+\sqrt{3}$ and $2+\sqrt{3}$
Question 201 Mark
Give an example of two irrational numbers whose:
Difference is an irrational number.
Difference is an irrational number.
Answer
View full question & answer→2 irrational numbers with difference an irrational number will be $3-\sqrt{5}$ and $3+\sqrt{5}.$
Question 211 Mark
Without actual division, find the following rational numbers are terminating decimals.
$\frac{5}{12}$
$\frac{5}{12}$
Answer
View full question & answer→$\frac{5}{12}$
Denominator of $\frac{5}{12}$ is $12$.
And,
$12 = 2^2 \times 3$
So, $12$ has a prime factor $3$, which is other than $2$ and $5$.
Thus, $\frac{5}{12}$ is not a terminating decimal.
Denominator of $\frac{5}{12}$ is $12$.
And,
$12 = 2^2 \times 3$
So, $12$ has a prime factor $3$, which is other than $2$ and $5$.
Thus, $\frac{5}{12}$ is not a terminating decimal.
Question 221 Mark
Rationalise $\frac{1}{\sqrt{3}+\sqrt{2}}.$
Answer
View full question & answer→$\frac{1}{\sqrt{3}+\sqrt{2}}=\frac{1}{\sqrt{3}+\sqrt{2}}\times\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}}$
$=\frac{\sqrt{3}-\sqrt{2}}{\big(\sqrt{3}\big)^2-\big(\sqrt{2}\big)^2}$
$=\frac{\sqrt{3}-\sqrt{2}}{3-2}$
$=\sqrt{3}-\sqrt{2}$
$=\frac{\sqrt{3}-\sqrt{2}}{\big(\sqrt{3}\big)^2-\big(\sqrt{2}\big)^2}$
$=\frac{\sqrt{3}-\sqrt{2}}{3-2}$
$=\sqrt{3}-\sqrt{2}$
Question 231 Mark
Represent the following rational numbers on the number line:
$-2.4$
$-2.4$
Answer
View full question & answer→$-2.4=\frac{-24}{10}=\frac{-12}{5}=-2\frac{2}{5}$
Question 241 Mark
Find two irrational numbers between 0.16 and 0.17.
Answer
View full question & answer→Two irrational numbers between 0.16 and 0.17 are as follows: 0.1611161111611111611111... and 0.169669666...
Question 251 Mark
Evaluate:
$\big(64\big)^{\frac{1}{6}}$
$\big(64\big)^{\frac{1}{6}}$
Answer
View full question & answer→$\big(64\big)^{\frac{1}{6}}=(2^6)^{\frac{1}{6}}=2^{\big(6\times\frac{1}{6}\big)}=2^1=2$
Question 261 Mark
Solve: $\big(3-\sqrt{11}\big)\big(3+\sqrt{11}\big).$
Answer
View full question & answer→$\big(3-\sqrt{11}\big)\big(3+\sqrt{11}\big)$
$=3^2-\big(\sqrt{11}\big)^2$
$=9-11$
$=-2$
$=3^2-\big(\sqrt{11}\big)^2$
$=9-11$
$=-2$
Question 271 Mark
Without actual division, find the following rational numbers are terminating decimals.
$\frac{7}{24}$
$\frac{7}{24}$
Answer
View full question & answer→$\frac{7}{24}$
Denominator of $\frac{7}{24}$ is $24.$
And,
$24 = 2^3 \times 3$
So, $24$ has a prime factor $3,$ which is other than $2$ and $5.$
Thus, $\frac{7}{24}$ is not a terminating decimal.
Denominator of $\frac{7}{24}$ is $24.$
And,
$24 = 2^3 \times 3$
So, $24$ has a prime factor $3,$ which is other than $2$ and $5.$
Thus, $\frac{7}{24}$ is not a terminating decimal.
Question 281 Mark
Evaluate:
$\big(64\big)^{-\frac{1}{2}}$
$\big(64\big)^{-\frac{1}{2}}$
Answer
View full question & answer→$\big(64\big)^{-\frac{1}{2}}=\frac{1}{\big(64\big)^{\frac{1}{2}}}=\frac{1}{\big(8^2\big)^{\frac{1}{2}}}=\frac{1}{\big(8\big)^{2\times\frac{1}{2}}}$
$=\frac{1}{8^1}=\frac{1}{8}$
$=\frac{1}{8^1}=\frac{1}{8}$
Question 291 Mark
Give an example of two irrational numbers whose:
Quotient is an irrational number.
Quotient is an irrational number.
Answer
View full question & answer→2 irrational numbers with quotient an irrational number will be $\sqrt{15}$ and $\sqrt{5}$
Question 301 Mark
Find an irrational number between 5 and 6.
Answer
View full question & answer→An irrational number between 5 and 6 $=\sqrt{5\times6}=\sqrt{30}$
Question 311 Mark
Give an example of two irrational numbers whose:
Product is an irrational number.
Product is an irrational number.
Answer
View full question & answer→2 irrational numbers with product an irrational number will be $6+\sqrt{3}$ and $7-\sqrt{3}$
Question 321 Mark
Give an example of two irrational numbers whose:
Sum is an irrational number.
Sum is an irrational number.
Answer
View full question & answer→2 irrational numbers with sum an irrational number $7+\sqrt{5}$ and $\sqrt{6}-8$
Question 331 Mark
Rationalise the denominator of the following:
$\frac{\sqrt{5}}{2\sqrt{3}}$
$\frac{\sqrt{5}}{2\sqrt{3}}$
Answer
View full question & answer→On multiplying the numerator and denominator of the given number by $\sqrt{3},$ we get
$\frac{\sqrt{5}}{2\sqrt{3}}=\frac{\sqrt{5}}{2\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{15}}{2\times3}=\frac{\sqrt{15}}{6}$
$\frac{\sqrt{5}}{2\sqrt{3}}=\frac{\sqrt{5}}{2\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{15}}{2\times3}=\frac{\sqrt{15}}{6}$
Question 341 Mark
Simplify $\Big(\frac{3125}{243}\Big)^{\frac{4}{5}}.$
Answer
View full question & answer→$\Big(\frac{3125}{243}\Big)^{\frac{4}{5}}$
$=\Big(\frac{5^5}{3^5}\Big)^{\frac{4}{5}}$
$=\Big(\frac{5}{3}\Big)^{5\times\frac{4}{5}}$
$=\Big(\frac{5}{3}\Big)^4$
$=\frac{625}{81}$
$=\Big(\frac{5^5}{3^5}\Big)^{\frac{4}{5}}$
$=\Big(\frac{5}{3}\Big)^{5\times\frac{4}{5}}$
$=\Big(\frac{5}{3}\Big)^4$
$=\frac{625}{81}$
Question 351 Mark
Classify the following number as rational or irrational. give reasons to support your answer.
$\sqrt{1.44}$
$\sqrt{1.44}$
Answer
View full question & answer→$\sqrt{1.44}=1.2$
So, it is rational.
So, it is rational.
Question 361 Mark
Simplify:
$(14641)^{0.25}$
$(14641)^{0.25}$
Answer
View full question & answer→$(14641)^{0.25}$
$=(14641)^{\frac{1}{4}}$
$=(11^4)^{\frac{1}{4}}$
$=11^{4\times\frac{1}{4}}$
$=11$
$=(14641)^{\frac{1}{4}}$
$=(11^4)^{\frac{1}{4}}$
$=11^{4\times\frac{1}{4}}$
$=11$
Question 371 Mark
Without actual division, find the following rational numbers are terminating decimals.
$\frac{13}{80}$
$\frac{13}{80}$
Answer
View full question & answer→$\frac{13}{80}$
Denominator of $\frac{13}{80}$ is $80.$
And,
$80 = 2^4 \times 5$
Therefore, $80$ has no other factors than $2$ and $5.$
Thus, $\frac{13}{80}$ is a terminating decimal.
Denominator of $\frac{13}{80}$ is $80.$
And,
$80 = 2^4 \times 5$
Therefore, $80$ has no other factors than $2$ and $5.$
Thus, $\frac{13}{80}$ is a terminating decimal.
Question 381 Mark
Simplify:
$\frac{5^{\frac{6}{7}}}{5^{\frac{2}{3}}}$
$\frac{5^{\frac{6}{7}}}{5^{\frac{2}{3}}}$
Answer
View full question & answer→$\frac{5^{\frac{6}{7}}}{5^{\frac{2}{3}}}=5^{\big(\frac{6}{7}-\frac{2}{3}\big)}$
$=5^{\big(\frac{18-14}{21}\big)}=5^{\frac{4}{21}}$
$=5^{\big(\frac{18-14}{21}\big)}=5^{\frac{4}{21}}$
Question 391 Mark
Simplify:
$2^\frac{2}{3}\times2^\frac{1}{5}$
$2^\frac{2}{3}\times2^\frac{1}{5}$
Answer
View full question & answer→$2^\frac{2}{3}\times2^\frac{1}{5}$
$=2^{\frac{2}{3}+\frac{1}{5}}$
$=2^{\frac{10+3}{15}}$
$=2^{\frac{13}{15}}$
$=2^{\frac{2}{3}+\frac{1}{5}}$
$=2^{\frac{10+3}{15}}$
$=2^{\frac{13}{15}}$
Question 401 Mark
Simplify:
$\frac{8^{\frac{1}{2}}}{8^{\frac{2}{3}}}$
$\frac{8^{\frac{1}{2}}}{8^{\frac{2}{3}}}$
Answer
View full question & answer→$\frac{8^{\frac{1}{2}}}{8^{\frac{2}{3}}}=8^{\big(\frac{1}{2}-\frac{2}{3}\big)}$
$=8^{\big(\frac{3-4}{6}\big)}=8^{\frac{-1}{6}}$
$=8^{\big(\frac{3-4}{6}\big)}=8^{\frac{-1}{6}}$
Question 411 Mark
Classify the following number as rational or irrational. give reasons to support your answer.
1.23232333...
1.23232333...
Answer
View full question & answer→1.23232333... is an irrational number because it is a non−terminating, non−repeating decimal.
Question 421 Mark
Simplify $\sqrt[4]{81\text{x}^8\text{y}^4\text{z}^{16}}.$
Answer
View full question & answer→$\sqrt[4]{81\text{x}^8\text{y}^4\text{z}^{16}}$
$=\sqrt[4]{3^4(\text{x}^2)^4\text{y}^4(\text{z}^4)^4}$
$=\sqrt[4]{(3\text{x}^2\text{y}\text{z}^4)^4}$
$=(3\text{x}^2\text{y}\text{z}^4)^{4\times\frac{1}{4}}$
$=3\text{x}^2\text{y}\text{z}^4$
$=\sqrt[4]{3^4(\text{x}^2)^4\text{y}^4(\text{z}^4)^4}$
$=\sqrt[4]{(3\text{x}^2\text{y}\text{z}^4)^4}$
$=(3\text{x}^2\text{y}\text{z}^4)^{4\times\frac{1}{4}}$
$=3\text{x}^2\text{y}\text{z}^4$
Question 431 Mark
Write the rationalising factor of the denominator in $\frac{1}{\sqrt{2}+\sqrt{3}}.$
Answer
View full question & answer→The rationalising factor of the denominator in $\frac{1}{\sqrt{2}+\sqrt{3}}$ is $\big(\sqrt{2}-\sqrt{3}\big).$
Question 441 Mark
Add:
$\Big(\frac{2}{3}\sqrt{7}-\frac{1}{2}\sqrt{2}+6\sqrt{11}\Big)$ and $\Big(\frac{1}{3}\sqrt{7}+\frac{3}{2}\sqrt{2}-\sqrt{11}\Big)$
$\Big(\frac{2}{3}\sqrt{7}-\frac{1}{2}\sqrt{2}+6\sqrt{11}\Big)$ and $\Big(\frac{1}{3}\sqrt{7}+\frac{3}{2}\sqrt{2}-\sqrt{11}\Big)$
Answer
View full question & answer→We have:
$\Big(\frac{2}{3}\sqrt{7}-\frac{1}{2}\sqrt{2}+6\sqrt{11}\Big)+\Big(\frac{1}{3}\sqrt{7}+\frac{3}{2}\sqrt{2}-\sqrt{11}\Big)$
$=\Big(\frac{2}{3}\sqrt{7}+\frac{1}{3}\sqrt{7}\Big)+\Big(-\frac{1}{2}\sqrt{2}+\frac{3}{2}\sqrt{2}\Big)+\big(6\sqrt{11}-\sqrt{11}\big)$
$=\Big(\frac{2}{3}+\frac{1}{3}\Big)\sqrt{7}+\Big(-\frac{1}{2}+\frac{3}{2}\Big)\sqrt{2}+(6-1)\sqrt{11}$
$=\sqrt{7}+\sqrt{2}+5\sqrt{11}$
$\Big(\frac{2}{3}\sqrt{7}-\frac{1}{2}\sqrt{2}+6\sqrt{11}\Big)+\Big(\frac{1}{3}\sqrt{7}+\frac{3}{2}\sqrt{2}-\sqrt{11}\Big)$
$=\Big(\frac{2}{3}\sqrt{7}+\frac{1}{3}\sqrt{7}\Big)+\Big(-\frac{1}{2}\sqrt{2}+\frac{3}{2}\sqrt{2}\Big)+\big(6\sqrt{11}-\sqrt{11}\big)$
$=\Big(\frac{2}{3}+\frac{1}{3}\Big)\sqrt{7}+\Big(-\frac{1}{2}+\frac{3}{2}\Big)\sqrt{2}+(6-1)\sqrt{11}$
$=\sqrt{7}+\sqrt{2}+5\sqrt{11}$
Question 451 Mark
Evaluate $\Big(\frac{81}{49}\Big)^{\frac{-3}{2}}.$
Answer
View full question & answer→$\Big(\frac{81}{49}\Big)^{\frac{-3}{2}}$
$=\Big(\frac{49}{81}\Big)^{\frac{3}{2}}$
$=\Big(\frac{7^2}{9^2}\Big)^{\frac{3}{2}}$
$=\Big(\frac{7}{9}\Big)^{2\times\frac{3}{2}}$
$=\Big(\frac{7}{9}\Big)^3$
$=\frac{343}{729}$
$=\Big(\frac{49}{81}\Big)^{\frac{3}{2}}$
$=\Big(\frac{7^2}{9^2}\Big)^{\frac{3}{2}}$
$=\Big(\frac{7}{9}\Big)^{2\times\frac{3}{2}}$
$=\Big(\frac{7}{9}\Big)^3$
$=\frac{343}{729}$
Question 461 Mark
Represent the following rational numbers on the number line:
$1.3$
$1.3$
Answer
View full question & answer→$1.3=\frac{13}{10}=1\frac{3}{10}$
Question 471 Mark
Simplify $\sqrt[4]{\sqrt[3]{\text{x}^2}}$ and express the result in the exponential form of x.
Answer
View full question & answer→$\sqrt[4]{\sqrt[3]{\text{x}^2}}$ $=\Big(\sqrt[3]{\text{x}^2}\Big)^\frac{1}{4}$ $=\big(\text{x}^2\big)^{\frac{1}{3}\times\frac{1}{4}}$ $=\text{x}^{2\times\frac{1}{12}}$ $=\text{x}^\frac{1}{6}$
Question 481 Mark
Simplify $(32)^\frac{1}{5}+(-7)^0+(64)^{\frac{1}{2}}.$
Answer
View full question & answer→$(32)^\frac{1}{5}+(-7)^0+(64)^{\frac{1}{2}}$
$=(2^5)^{\frac{1}{5}}+1+(8^2)^{\frac{1}{2}}$
$=2^{5\times\frac{1}{5}}+1+8^{2\times\frac{1}{2}}$
$=2+1+8$
$=11$
$=(2^5)^{\frac{1}{5}}+1+(8^2)^{\frac{1}{2}}$
$=2^{5\times\frac{1}{5}}+1+8^{2\times\frac{1}{2}}$
$=2+1+8$
$=11$
Question 491 Mark
If $a = 1, b = 2$ then find the value of $(a^b + b^a)^{-1}.$
Answer
View full question & answer→Given, $a = 1$ and $b = 2$
$\therefore(\text{a}^{\text{b}}+\text{b}^{\text{a}})^{-1}$
$=\frac{1}{\text{a}^{\text{b}}+\text{b}^{\text{a}}}$
$=\frac{1}{1^2+2^1}$
$=\frac{1}{1+2}$
$=\frac{1}{3}$
$\therefore(\text{a}^{\text{b}}+\text{b}^{\text{a}})^{-1}$
$=\frac{1}{\text{a}^{\text{b}}+\text{b}^{\text{a}}}$
$=\frac{1}{1^2+2^1}$
$=\frac{1}{1+2}$
$=\frac{1}{3}$
Question 501 Mark
Simplify $6\sqrt{36}+5\sqrt{12}$
Answer
View full question & answer→$6\sqrt{3}+5\sqrt{12}$
$=6\sqrt{3}+5\sqrt{4\times3}$
$=6\sqrt{3}+5\times2\sqrt{3}$
$=6\sqrt{3}+10\sqrt{3}$
$=16\sqrt{3}$
$=6\sqrt{3}+5\sqrt{4\times3}$
$=6\sqrt{3}+5\times2\sqrt{3}$
$=6\sqrt{3}+10\sqrt{3}$
$=16\sqrt{3}$